According to the analysis in Section 3.2, the change of the AE ring counts can characterize the entire process of instability and destruction of the strip and column CGBBs. In addition, according to the analysis in Section 3.3, before the peak stress, with the increase of the stress, the resistivity also increases gradually. When the peak stress is reached, the resistivity of the CGBB suddenly increases to the maximum value. The above features can be regarded as the characteristic points of the instability of the CGBB to realize the stability monitoring of the strip and column CGBBs in constructional backfill mining. In the process of monitoring the stability of the CGBB, the accumulative AE ring counts and resistivity could be used to judge the stress, strain, and damage state of the CGBB, to realize the stability monitoring of the strip and column CGBBs in the goaf. Therefore, it is necessary to establish the relationship between cumulative ring counts, resistivity and stress, strain, and damage.
The damage constitutive model of the rock based on the cumulative AE ring counts can be established (Wu et al. 2015). This uniaxial compression test adopts the displacement loading method, and the relationship between strain and time is linear. Therefore, the relationship between strain \(\varepsilon\) and time is:
$$\varepsilon {\text{=}}kt+{\varepsilon _0}$$
7
Where is the strain rate; \({\varepsilon _0}\) is the initial strain.
As shown in Fig. 5(b) and Fig. 6(b), the relationship between cumulative AE ring counts and time can be expressed by the index function:
$$N={A_1}\exp ( - \frac{t}{B})+{A_2}$$
8
Where, , \({A_2}\),and are determined by the fitting results of experimental results.
The relationship between cumulative AE ring counts and strain \(\varepsilon\) can be obtained by combining the formula (7) and (8).
$$\varepsilon {\text{=}}{\varepsilon _0}{\text{-}}Bk\ln (\frac{{N - {A_2}}}{{{A_1}}})$$
9
Therefore, according to formula (3) and formula (9), the damage variable can be defined as:
$$D=1 - \exp \left\{ { - \frac{1}{\alpha }{{\left[ {\left. {{\varepsilon _0} - Bk\ln \left( {\frac{{N - {A_2}}}{{{A_1}}}} \right)} \right]} \right.}^m}} \right\}$$
10
According to formula (2) and formula (10), the relationship between cumulative AE ring counts and stress \(\sigma\) of the CGBB is:
$$\sigma =E\left[ {\left. {{\varepsilon _0} - Bk\ln \left( {\frac{{N - {A_2}}}{{{A_1}}}} \right)} \right]} \right.\exp \left\{ { - \frac{1}{\alpha }{{\left[ {\left. {{\varepsilon _0} - Bk\ln \left( {\frac{{N - {A_2}}}{{{A_1}}}} \right)} \right]} \right.}^m}} \right\}$$
11
The model parameters were fitted by the column CGBB, and the fitting results are shown in Table 3. The relationship between the stress, strain, and damage factors of the theory is plotted in Fig. 13a.
Similarly, as shown in Fig. 7, the relationship of the resistivity \(\rho\) and time can be expressed as:
$$\rho ={D_1}\exp ( - \frac{t}{{{H_1}}})+F$$
12
Where \({D_1}\), \({H_1}\), are determined by the fitting results of experimental data.
Combination of the formula (7) and formula (12), the relationship between resistivity \(\rho\) and strain \(\varepsilon\) is:
$$\varepsilon {\text{=}}{\varepsilon _0}{\text{-}}{H_1}k\ln (\frac{{\rho - F}}{{{D_1}}})$$
13
Combination of the formula (13) and formula (3), the relationship between damage variable and resistivity and the relationship between stress and resistivity are:
$$D=1 - \exp \left\{ { - \frac{1}{\alpha }\left[ {{\varepsilon _0} - {H_1}k\ln {{(\frac{{\rho - F}}{{{D_1}}})}^m}} \right]} \right\}$$
14
Combination of the formula (14) and formula (2), the relationship between stress \(\sigma\) and resistivity \(\rho\) can be expressed as:
$$\sigma =E\left[ {{\varepsilon _0} - {H_1}k\ln (\frac{{\rho - F}}{{{D_1}}})} \right]\exp \left\{ { - \frac{1}{\alpha }{{\left[ {{\varepsilon _0} - {H_1}k\ln (\frac{{\rho - F}}{{{D_1}}})} \right]}^m}} \right\}$$
15
The horizontal resistivity monitoring result of the strip CGBB specimen was selected to verify the stability monitoring model of resistivity. Table 3 shows the calculation parameters. This model calculation results and experimental data are shown in Fig. 13b.
It can be seen from Fig. 13 that the experimental data and the model calculation results are in good agreement as a whole, while the consistency is not very high in the compaction stage of the strip and column CGBBs. Therefore, these models can be used in the stability monitoring of CGBB in constructional backfill mining.
Table 3
The parameters of acoustic emission cumulative ring counts model and resistivity model.
Acoustic emission cumulative ring
|
\({\sigma _P}\)/MPa
|
\({\varepsilon _P}\)
|
/MPa
|
\(\alpha\)
|
|
|
\({\varepsilon _0}\)
|
\({A_1}\)
|
|
\({A_2}\)
|
Parameter value
|
5.1223
|
0.00992
|
1137.8
|
3.686E-3
|
1.265
|
8.886E-5
|
1.869E-4
|
-4102.72
|
39.66
|
3996.32
|
Resistivity
|
\({\sigma _P}\)/MPa
|
\({\varepsilon _P}\)
|
/MPa
|
\(\alpha\)
|
|
|
\({\varepsilon _0}\)
|
\({D_1}\)
|
\({H_1}\)
|
|
Parameter value
|
5.854
|
0.0298
|
373.7
|
6.594E-3
|
1.555
|
1.3324E-4
|
0.0024
|
11.392
|
-34.58
|
26.193
|